In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers.^[a] Equivalently, a set is countable if there exists an injective function from it into the natural numbers; this means that each element in the set may be associated to a unique natural number, or that the elements of the set can be counted one at a time, although the counting may never finish due to an infinite number of elements.

In more technical terms, assuming the axiom of countable choice, a set is countable if its cardinality (the number of elements of the set) is not greater than that of the natural numbers. A countable set that is not finite is said to be countably infinite.

The concept is attributed to Georg Cantor, who proved the existence of uncountable sets, that is, sets that are not countable; for example the set of the real numbers.

A note on terminology

Although the terms "countable" and "countably infinite" as defined here are quite common, the terminology is not universal.^[1] An alternative style uses countable to mean what is here called countably infinite, and at most countable to mean what is here called countable.^[2][3]

The terms enumerable^[4] and denumerable^[5][6] may also be used, e.g. referring to countable and countably infinite respectively,^[7] definitions vary and care is needed respecting the difference with recursively enumerable.^[8]

Definition

A set ${\displaystyle S}$ is countable if:

• Its cardinality ${\displaystyle |S|}$ is less than or equal to ${\displaystyle \aleph _{0}}$ (aleph-null), the cardinality of the set of natural numbers ${\displaystyle \mathbb {N} }$.^[9]
• There exists an injective function from ${\displaystyle S}$ to ${\displaystyle \mathbb {N} }$.^[10][11]
• ${\displaystyle S}$ is empty or there exists a surjective function from ${\displaystyle \mathbb {N} }$ to ${\displaystyle S}$.^[11]
• There exists a bijective mapping between ${\displaystyle S}$ and a subset of ${\displaystyle \mathbb {N} }$.^[12]
• ${\displaystyle S}$ is either finite (${\displaystyle |S|<\aleph _{0}}$) or countably infinite.^[5]

All of these definitions are equivalent.

A set ${\displaystyle S}$ is countably infinite if:

• Its cardinality ${\displaystyle |S|}$ is exactly ${\displaystyle \aleph _{0}}$.^[9]
• There is an injective and surjective (and therefore bijective) mapping between ${\displaystyle S}$ and ${\displaystyle \mathbb {N} }$.
• ${\displaystyle S}$ has a one-to-one correspondence with ${\displaystyle \mathbb {N} }$.^[13]
• The elements of ${\displaystyle S}$ can be arranged in an infinite sequence ${\displaystyle a_{0},a_{1},a_{2},\ldots }$, where ${\displaystyle a_{i}}$ is distinct from ${\displaystyle a_{j}}$ for ${\displaystyle i\neq j}$ and every element of ${\displaystyle S}$ is listed.^[14][15]

A set is uncountable if it is not countable, i.e. its cardinality is greater than ${\displaystyle \aleph _{0}}$.^[9]

History

In 1874, in his first set theory article, Cantor proved that the set of real numbers is uncountable, thus showing that not all infinite sets are countable.^[16] In 1878, he used one-to-one correspondences to define and compare cardinalities.^[17] In 1883, he extended the natural numbers with his infinite ordinals, and used sets of ordinals to produce an infinity of sets having different infinite cardinalities.^[18]

Introduction

A set is a collection of elements, and may be described in many ways. One way is simply to list all of its elements; for example, the set consisting of the integers 3, 4, and 5 may be denoted ${\displaystyle \{3,4,5\}}$, called roster form.^[19] This is only effective for small sets, however; for larger sets, this would be time-consuming and error-prone. Instead of listing every single element, sometimes an ellipsis ("...") is used to represent many elements between the starting element and the end element in a set, if the writer believes that the reader can easily guess what ... represents; for example, ${\displaystyle \{1,2,3,\dots ,100\}}$ presumably denotes the set of integers from 1 to 100. Even in this case, however, it is still possible to list all the elements, because the number of elements in the set is finite. If we number the elements of the set 1, 2, and so on, up to ${\displaystyle n}$, this gives us the usual definition of "sets of size ${\displaystyle n}$".

Some sets are infinite; these sets have more than ${\displaystyle n}$ elements where ${\displaystyle n}$ is any integer that can be specified. (No matter how large the specified integer ${\displaystyle n}$ is, such as ${\displaystyle n=10^{1000}}$, infinite sets have more than ${\displaystyle n}$ elements.) For example, the set of natural numbers, denotable by ${\displaystyle \{0,1,2,3,4,5,\dots \}}$,^[a] has infinitely many elements, and we cannot use any natural number to give its size. It might seem natural to divide the sets into different classes: put all the sets containing one element together; all the sets containing two elements together; ...; finally, put together all infinite sets and consider them as having the same size. This view works well for countably infinite sets and was the prevailing assumption before Georg Cantor's work. For example, there are infinitely many odd integers, infinitely many even integers, and also infinitely many integers overall. We can consider all these sets to have the same "size" because we can arrange things such that, for every integer, there is a distinct even integer:

or, more generally, ${\displaystyle n\rightarrow 2n}$ (see picture). What we have done here is arrange the integers and the even integers into a one-to-one correspondence (or bijection), which is a function that maps between two sets such that each element of each set corresponds to a single element in the other set. This mathematical notion of "size", cardinality, is that two sets are of the same size if and only if there is a bijection between them. We call all sets that are in one-to-one correspondence with the integers countably infinite and say they have cardinality ${\displaystyle \aleph _{0}}$.

Georg Cantor showed that not all infinite sets are countably infinite. For example, the real numbers cannot be put into one-to-one correspondence with the natural numbers (non-negative integers). The set of real numbers has a greater cardinality than the set of natural numbers and is said to be uncountable.

Formal overview

By definition, a set ${\displaystyle S}$ is countable if there exists a bijection between ${\displaystyle S}$ and a subset of the natural numbers ${\displaystyle \mathbb {N} =\{0,1,2,\dots \}}$. For example, define the correspondence

Since every element of ${\displaystyle S=\{a,b,c\}}$ is paired with precisely one element of ${\displaystyle \{1,2,3\}}$, and vice versa, this defines a bijection, and shows that ${\displaystyle S}$ is countable. Similarly we can show all finite sets are countable.

As for the case of infinite sets, a set ${\displaystyle S}$ is countably infinite if there is a bijection between ${\displaystyle S}$ and all of ${\displaystyle \mathbb {N} }$. As examples, consider the sets ${\displaystyle A=\{1,2,3,\dots \}}$, the set of positive integers, and ${\displaystyle B=\{0,2,4,6,\dots \}}$, the set of even integers. We can show these sets are countably infinite by exhibiting a bijection to the natural numbers. This can be achieved using the assignments ${\displaystyle n\leftrightarrow n+1}$ and ${\displaystyle n\leftrightarrow 2n}$, so that